Why is blender exporting face normals instead of vertex normals?
If you're just switching between Shade Flat and Shade Smooth you should see a difference in the viewport and in the exported .obj file. If you have Auto-Smooth enabled, then this will influence the normals and could make the result identical to Shade Flat if the Auto-Smooth Angle is lower than the 90° angles in the cube.

Shade Flat

# Blender v2.81 (sub 16) OBJ File: ''
# www.blender.org
o Cube
v 1.000000 1.000000 -1.000000
v 1.000000 -1.000000 -1.000000
v 1.000000 1.000000 1.000000
v 1.000000 -1.000000 1.000000
v -1.000000 1.000000 -1.000000
v -1.000000 -1.000000 -1.000000
v -1.000000 1.000000 1.000000
v -1.000000 -1.000000 1.000000
vt 0.625000 0.500000
vt 0.875000 0.500000
vt 0.875000 0.750000
vt 0.625000 0.750000
vt 0.375000 0.750000
vt 0.625000 1.000000
vt 0.375000 1.000000
vt 0.375000 0.000000
vt 0.625000 0.000000
vt 0.625000 0.250000
vt 0.375000 0.250000
vt 0.125000 0.500000
vt 0.375000 0.500000
vt 0.125000 0.750000
vn 0.0000 1.0000 0.0000
vn 0.0000 0.0000 1.0000
vn -1.0000 0.0000 0.0000
vn 0.0000 -1.0000 0.0000
vn 1.0000 0.0000 0.0000
vn 0.0000 0.0000 -1.0000
s off
f 1/1/1 5/2/1 7/3/1 3/4/1
f 4/5/2 3/4/2 7/6/2 8/7/2
f 8/8/3 7/9/3 5/10/3 6/11/3
f 6/12/4 2/13/4 4/5/4 8/14/4
f 2/13/5 1/1/5 3/4/5 4/5/5
f 6/11/6 5/10/6 1/1/6 2/13/6




Shade Smooth

# Blender v2.81 (sub 16) OBJ File: ''
# www.blender.org
o Cube
v 1.000000 1.000000 -1.000000
v 1.000000 -1.000000 -1.000000
v 1.000000 1.000000 1.000000
v 1.000000 -1.000000 1.000000
v -1.000000 1.000000 -1.000000
v -1.000000 -1.000000 -1.000000
v -1.000000 1.000000 1.000000
v -1.000000 -1.000000 1.000000
vt 0.625000 0.500000
vt 0.875000 0.500000
vt 0.875000 0.750000
vt 0.625000 0.750000
vt 0.375000 0.750000
vt 0.625000 1.000000
vt 0.375000 1.000000
vt 0.375000 0.000000
vt 0.625000 0.000000
vt 0.625000 0.250000
vt 0.375000 0.250000
vt 0.125000 0.500000
vt 0.375000 0.500000
vt 0.125000 0.750000
vn 0.5773 0.5773 -0.5773
vn -0.5773 0.5773 -0.5773
vn -0.5773 0.5773 0.5773
vn 0.5773 0.5773 0.5773
vn 0.5773 -0.5773 0.5773
vn -0.5773 -0.5773 0.5773
vn -0.5773 -0.5773 -0.5773
vn 0.5773 -0.5773 -0.5773
s 1
f 1/1/1 5/2/2 7/3/3 3/4/4
f 4/5/5 3/4/4 7/6/3 8/7/6
f 8/8/6 7/9/3 5/10/2 6/11/7
f 6/12/7 2/13/8 4/5/5 8/14/6
f 2/13/8 1/1/1 3/4/4 4/5/5
f 6/11/7 5/10/2 1/1/1 2/13/8